Topics in Integration

  • J. Stoer
  • R. Bulirsch

Abstract

Calculating the definite integral of a given real function f (x),
$$\int_a^b {f(x)dx} $$
, is a classic problem. For some simple integrands f (x), the indefinite integral
$$\int_a^b {f(x)dx} = F(x),\quad F'(x) = f(x),$$
can be obtained in closed form as an algebraic expression in x and well-known transcendental functions of x.

Keywords

Asymptotic Expansion Orthogonal Polynomial Step Length Extrapolation Method Hermite Interpolation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References for Chapter 3

  1. Abramowitz, M., Stegun, I. A.: Handbook of Mathematical Functions. National Bureau of Standards, Applied Mathematics Series 55, Washington, D.C.: U.S. Government Printing Office 1964, 6th printing 1967.Google Scholar
  2. Bauer, F. L., Rutishauser, H., Stiefel, E.: New aspects in numerical quadrature. Proc. of Symposia in Applied Mathematics 15, 199–218, Amer. Math. Soc. 1963.MathSciNetCrossRefGoogle Scholar
  3. Bulirsch, R.: Bemerkungen zur Romberg-Integration. Numer. Math. 6, 6–16 (1964).MathSciNetMATHCrossRefGoogle Scholar
  4. Bulirsch, R., Stoer, J.: Fehlerabschätzungen und Extrapolation mit rationalen Funktionen bei Verfahren vom Richardson-Typus. Numer. Math. 6, 413–427 (1964).MathSciNetMATHCrossRefGoogle Scholar
  5. Bulirsch, R., Stoer, J.: Numerical quadrature by extrapolation. Numer. Math. 271–278 (1967).Google Scholar
  6. Davis, P. J.: Interpolation and Approximation. New York: Blaisdell 1963, 2nd printing 1965.MATHGoogle Scholar
  7. Davis, P. J., Rabinowitz, P.: Methods of Numerical Integration. New York: Academic Press 1975.MATHGoogle Scholar
  8. Erdelyi, A.: Asymptotic Expansions. New York: Dover 1956.MATHGoogle Scholar
  9. Gautschi, W.: Construction of Gauss-Christoffel quadrature formulas. Math. Comp. 22, 251–270 (1968).MathSciNetMATHCrossRefGoogle Scholar
  10. Gautschi, W.: On the construction of Gaussian quadrature rules from modified moments. Math. Comput. 24, 245–260 (1970).MathSciNetMATHGoogle Scholar
  11. Golub, G. H., Welsch, J. H.: Calculation of Gauss quadrature rules. Math. Comput. 23, 221–230 (1969).MathSciNetMATHCrossRefGoogle Scholar
  12. Gröbner, W., Hofreiter, N.: Integraltafel, 2 vols. Berlin: Springer Verlag 1961.Google Scholar
  13. Kronrod, A. S.: Nodes and Weights of Quadrature Formulas. Authorized translation from the Russian. New York: Consultants Bureau 1965.MATHGoogle Scholar
  14. Henrici, P.: Elements of Numerical Analysis. New York: Wiley 1964.MATHGoogle Scholar
  15. Olver, F. W. J.: Asymptotics and Special Functions. New York: Academic Press 1974.Google Scholar
  16. Romberg, W.: Vereinfachte numerische Integration. Det. Kong. Norske Videnskabers Selskab Forhandlinger 28, Nr. 7, Trondheim 1955.Google Scholar
  17. Schoenberg, I. J.: Monosplines and quadrature formulae. In: Theory and Applications of Spline Functions. Edited by T. N. E. Greville. 157–207. New York: Academic Press 1969.Google Scholar
  18. Steffensen, J. F.: Interpolation (1927) 2nd edition. New York: Chelsea 1950.MATHGoogle Scholar
  19. Stroud, A. H., Secrest, D.: Gaussian Quadrature Formulas. Englewood Cliffs, N.J.: Prentice-Hall 1966.MATHGoogle Scholar
  20. Szegö, G.: Orthogonal Polynomials. New York: Amer. Math. Soc. 1959.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1980

Authors and Affiliations

  • J. Stoer
    • 1
  • R. Bulirsch
    • 2
  1. 1.Institut für Angewandte MathematikUniversität Würzburg am HublandWürzburgFederal Republic of Germany
  2. 2.Institut für MathematikTechnische UniversitätMünchenFederal Republic of Germany

Personalised recommendations